The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: gauss

Euclidean geometry is the geometry that everyone learns and uses throughout Middle School and High School. In general, geometry is the study of figures, such as points, lines and circles in space. Euclidean geometry is specifically any geometry in which all of Euclid’s postulates and axioms hold. Axioms and postulates are the beginning of reasoning, they are simple statements that are believed to be true without proof. Assuming Euclid’s axioms and postulates found in his book Elements, the rest of Euclid’s classical geometry could be deduced. However, Euclid’s fifth postulate, the parallel postulate, was disconcerting because it was lengthy compared to the rest and not necessarily self evident. Many other ancient mathematicians were dissatisfied with Euclid’s fifth postulate. They thought that it was presumptuous and tried to prove it using lesser axioms or replace it altogether with something they thought to be more intuitive. But their proofs always included an assumption equivalent to the parallel postulate, so for centuries the postulate was assumed to be true.

Centuries passed and the postulate remained unproven; however, development to understand Euclid’s postulate continued into the eighteenth century. Perhaps the most well-known equivalent to the parallel postulate is Playfair’s Axiom, which states “through any point in the plane, there is at most one straight line parallel to a given straight line.” Arguably one of the most influential mathematicians, Carl Friedrich Gauss became interested in proving Euclid’s fifth postulate. After attempting to prove the postulate, he instead took Playfair’s Axiom and altered it, creating a completely new postulate. Gauss’ new postulate stated “Through a given point not on a line, there are at least two lines parallel to the given line through the given point.” With this Gauss had unearthed a completely new space that today is called hyperbolic geometry. However, he chose not to publish any of his results, wishing not to get caught up in any political strife. The work was later published by Johann Bolyai and Nikolay Lobachevesky, who both had academic ties to Gauss.

Shortly after this discovery another type of Non-Euclidean geometry was discovered by Gauss’ student Georg Friedrich Bernhard Riemann. Riemann looked at what would happen when you altered Playfiar’s Axiom in the opposite direction than Gauss. Riemann’s alternate postulate is stated as follows, “through a given point not on a line, there exist no lines parallel to the line through the given point.” With this, what is known as elliptical or spherical geometry was discovered.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry provides a somewhat simpler model then hyperbolic geometry. Anyone who has ever attempted to wrap a basketball in paper understands that there are some discrepancies between the two surfaces. For instance, triangles behave quite differently than they do in Euclidean geometry. In hyperbolic space, the angles of a triangle add up to less than 180 degrees, and in spherical space, they add up to more than 180 degrees. Spherical geometry also has other inconsistencies with Euclid’s initial assumptions other than the parallel postulate. In Leonard Mlodinow’s book Euclid’s Window, the author describes how Riemann’s space was inconsistent with other of Euclid’s postulates. For instance, Euclid’s second postulate states that “any line segment can be extended indefinitely in either direction.” In spherical space this is not true; the lines, or great circles, have a limit to their space, namely two pi times the radius of the sphere. Mlodinow describes how Riemann saw this postulate as “only necessary to guarantee that the lines had no bounds, which is true of the great circles.” Also, Euclid’s first postulate became less clear, “Given any two points, a line segment can be drawn with those points as its endpoints.” This postulate can be used to easily describe whether a point is between two other points. However, on the globe, choosing two points on the equator such as Ecuador and Indonesia it is difficult to say whether a third point, Kenya, is “between” them. The problem is that there are two ways to connect the points, one passing over North America and another passing over Africa.

For much of our day to day lives Euclidean geometry works great, because on a local scale we appear to live on a flat world. I can go to a soccer field and trust that it will take four 90 degree turns to walk around the perimeter, or that the Pythagorean theorem will work to describe the path between opposite corners. But looking at a larger scale, the surface that we live on is spherical and has different properties than the flat plane. It is interesting to see how Gauss and Riemann, going against the grain of conventional mathematics, led to new and vast fields of undiscovered mathematics. To me, this shows how mathematics is just as much an experimental science as physics or engineering. These new discoveries of mathematical spaces made possible Einstein’s physical description of the space in which we live. Mlodinow closes his section on Gauss and Riemann saying, “though thoroughly remodeled, geometry continued to be the window to understanding our universe.” Even though the properties of these new geometries differ from classic Euclidean geometry and may have more or less practical use, they are just as important. From Euclid up until Gauss, mathematics was largely pragmatic, but the discovery of these new geometries highlights how math can be appreciated for its own sake.

It’s kind of embarrassing to admit this as a 24-year-old college student, but I have an imaginary friend. His name is √(-1). He doesn’t keep it as real as some of my other friends, but he’s always my main dude in some of the more complex friendships I get involved in. He’s got a pretty interesting life story, and he’s exceptionally old and wise (imaginary friends never die). He was birthed by the Greek mathematician Heron of Alexandria in the first century A.D. He had a rough childhood (neglected and completely ignored by his father), partially because no one understood him. People ignored him, calling him useless. His name was even meant to be derogatory at one point.4 √(-1) met Gerolamo Cardano (coined the term “imaginary”), his first true friend, in 1637. After Cardano passed away, √(-1) was lonely until he was accepted into a new circle of friends during the 1700 and 1800s: Leonhard Euler, Carl Gauss, and Caspar Wessel. Although his new friends were a little on the nerdy side, they helped restore his personal image – eventually helping him to become a internationally known, and well respected celebrity.

Image: Matheepan Panchalingam, via Flickr.

Okay, enough with my weird metaphorical stories. I’m here to tell you all about imaginary numbers. Now although the previous paragraph was strange and pretty corny, a lot of truth resides within. Heron of Alexandria was the first known person to have encountered imaginary numbers, back in the first century A.D.¹ He has been called a Greek mathematician, but interestingly enough, we now believe he may have actually been Egyptian.¹ So, how did Heron, the father of imaginary numbers, make such an important discovery? Well, he was trying to find the volume of a frustum of a pyramid with a square base. Specifically a pyramid with a side of the lower base is 28, of the upper 4, and the edge 15. Now the formula to solve this problem, as noted in the Stereometria of Heron of Alexandria is:

Using a = 28, b = 4, and c = 15, Heron ultimately reduced this problem to equal √(81 – 144). However instead of writing h = √(-63), he wrote h = √(63).¹ Now whether Heron did this intentionally or not, we’ll never know. What we do know is this: No other scholar even bothered to take notice of these imaginary numbers for at least a thousand years. Believe it or not, from here it took another five hundred years for scholars to start taking imaginary numbers seriously.¹

The Italian engineer and architect Rafael Bombelli started searching for solutions to cubic equations in 1572, and during that time, he started defining some guidelines for using imaginary numbers.5 At this point, most mathematicians didn’t want to believe these numbers were significant, or that they even existed. Bombelli’s “wild idea” – that you could multiply imaginary numbers and end up with a real solution, didn’t sit well with fellow scholars. In fact, Bombelli didn’t even think imaginary numbers could ever have a true purpose, he just viewed them as a useful artifact to solve his equations. Later on, along came Leonhard Euler, a math rockstar in many people’s eyes. Euler really takes the cake here for noticing one of math’s more bizarre imaginary (in a math sense) equations. He showed the world how closely i, e, and π are related and further convinced the math community that i was relevant:2 Today, this is known as Euler’s identity.

eiπ = cosπ +i(sinπ)

Which reduces to:

eiπ = -1 + 0i

In simpler terms:

eiπ = -1

In 1842 Carl Gauss did the impossible. He actually defined imaginary numbers (although Cardano coined the term “imaginary”.).Later on, once the reality of i was more widely accepted, our understanding mathematics expanded greatly. Euler, Gauss, and Wessel (1700s and 1800s) utilized imaginary numbers and led to the creation of complex numbers, which have a vital importance in math/science today.

Complex Numbers:

Instead of describing numbers and algebraic equations as points on a line, complex numbers and equations become points on a plane. Numbers are two dimensional – just like the integer “1″ is the unit distance on the axis of the “real” numbers, “i ” is the unit distance on the axis of the “imaginary” numbers. This results (in general) in complex numbers: They consist of two components, defining their position relative to those two axes. We generally write them as “a + bi” where “a” is the real component, and “b” is the imaginary component.4

A basic graph showing real and imaginary points making up a complex line. Image: Public Domain.

Let me simplify this for you – What this means is that every polynomial equation has roots. Specifically, a polynomial equation in “x” with maximum exponent “n” will always have exactly “n” complex roots.

Me, myself, and i :

Imaginary numbers are more prevalent in the world than you’d think. Without the electronic circuit theories conceived from imaginary numbers, I wouldn’t be typing this blog post, and more importantly, you wouldn’t be reading it. When we plug something into an electrical outlet, we just expect our electronic devices to receive power. In fact, I bet you plugged in your laptop charger today and didn’t even think twice about it. What does exactly does this AC power have to do with imaginary numbers? Everything.

Voltage can be viewed as the amount of force pushing the current through your laptop charger and ultimately into your laptop’s battery. Voltage is a complex number. In fact, if you have a voltage of 110 volts AC at 60 hz (US Standard), that means is that the voltage is a number of magnitude 110. If you were to plot the “real” voltage on a graph with time on the X axis and voltage of the Y, you’d see a basic sine wave. Take the graph below for example:

Basic sine wave showing only real voltage. Image: Public Domain.

If you decided to put your key in the outlet when the voltage was supposedly zero on the real portion of that graph, you’d still get electrocuted. That’s right, even though real voltage is zero here, “imaginary” electricity can give you quite a shock.3 Take the moment marked t1 on the above graph for example. The voltage at time t1 on the complex plane is a point at 110 on the real axis. At time t2, the voltage on the “real” axis is zero – but on the imaginary axis it’s 110. In fact, the magnitude of the voltage is always constant at 110 volts, and this is an important fact to know.5 All our modern day electronics account for these weird voltage patterns, and strangely enough this imaginary unit is represented with a “j ” rather than an i (“i ” is used to represent current)!3

All in all, you can thank Bombelli and Euler for introducing imaginary numbers. Not only do they make math more confusing, but they also have many crucial applications in the world around us today.

The common analogy used to describe modular arithmetic is fairly simple. All one has to do is look at an analog clock. For example, if it’s 11 AM and you want to know what time it will be in four hours, we instinctively know the answer is 3 PM. This is modular arithmetic, i.e. 11+4 = 3 mod 12. This is an important concept in the technology driven world we live in. Any time a product is purchased on the internet, cryptography comes into play. The remainder of this paper (pun most definitely intended) will describe how ancient modular arithmetic plays a very important role in today’s society.

History of modular arithmetic

The first known publication of modular arithmetic was in the 3rd century B.C.E, in the book Elements, written by Euclid. Within his book, he not only formalized the fundamentals of arithmetic, but also proved it. In what is known as Euclids Lemma, he states that if a prime number divides the product of two different numbers (x and y), then the prime number must also divide one of the numbers (either x or y), but it could also be both. Between the 3rd and 5th centuries a paper publish by Sun Tzu describes a modular arithmetic process known as the Chinese remainder theorem. This theorem is essentially the basis for modern RSA encryption schemes that are present on every banking/e-commerce website. It uses a congruent set of keys to produce the same numerical value. Imagine if there was a lock on a door that two differently cut keys could unlock and open, this is essentially how Chinese remainder theorem works.

The modular arithmetic that we use today was discovered by Carl Friedrich Gauss in 1801.

Gauss is famous for numerous discoveries across a wide variety of fields in science and mathematics. Gauss’s proposition, from his book Disquisitiones Arithmeticae, defines modular arithmetic by saying that any integer N belongs to a single residue-class when divided by a number M. The residue-class is represented by the remainder, which can be from 0 to M-1. The remainder is obtained by dividing N by M. Given this fact, Gauss notices that two numbers that differ by a multiple of M are in the same residue-class. He then discusses the role of negative numbers in modular arithmetic. The following is an excerpt from his book:

“The modulus m is usually positive, but there’s no great difficulty in allowing negative moduli (classes modulo m and -m are the same). For a zero modulus, there would be infinitely many residue classes, each containing only one element. [This need not be disallowed.]”

Modular Arithmetic’s Role Today

RSA encryption is named after those who invented it, Ron Rivest, Adi Shamir, and Leonard Adleman (RSA is obtained from their last names). RSA is the process by which information can be passed between two parties without another individual being able to intercept the message. Burt Kaliski has been one of the major contributors to RSA encryption since the 1980’s. I would like to start off with a passage from Burt Kaliski’s paper titled “The Mathematics of the RSA Public-Key Cryptosystem”:

“Sensitive data exchanged between a user and a Web site needs to be encrypted to prevent it from being disclosed to or modified by unauthorized parties. The encryption must be done in such a way that decryption is only possible with knowledge of a secret decryption key. The decryption key should only be known by authorized parties.”

This is a high level description of how RSA encryption works. It is also called public-key encryption, because anyone can obtain a copy of the encryption key it is publically available, but the decryption key cannot be obtained. This makes RSA encryption a secure way of passing data between an individual and a web site.

Simplified view of RSA encryption. Public Domain.

Performing this calculation (encrypting and decrypting text) is fairly simple. With a basic understanding of modular arithmetic it can be accomplished. First a public and private key must be produced by following the steps below:

Generate large prime numbers, p and q (these should be hundreds of digits)

Compute the modulus n, n = p×q

Compute the totient, totient = (p-1)×(q-1)

Choose an “e” > 1 that is co-prime to the totient

Choose a “d” such that d×e = 1 mod totient

Once those steps have been completed, a public key (n, e) and a private key (n, d) have been generated. The public key can be distributed to anyone, but the private key must be kept safe. It’s easy to see that without the modular arithmetic this algorithm would be easy to discern. One could generate pairs of random numbers until a pair is found that when multiplied together, would equal the modulus n found in step two above. From there, it would be easy to find all numbers co-prime to the totient in step three. Modular arithmetic then comes into play, because it allows infinite pairs of numbers to satisfy the constraint listed in step five, but it would not allow the user to decrypt the message. In other words, 11+4 = 3 mod 12, but also 11+16 = 3 mod 12. This makes it impossible to determine what the original number was (it could be 4 or it could be 16, or any other multiple of 12).

Once the keys have been generated it is easy to encrypt and decrypt text. To encrypt a message “m,” given the public key (n,e) generated above:

C = me mod n

“C” is then the encrypted message that gets passed to the other party.

To decrypt the message “C” created above, all that is required is the inverse of the operation to encrypt:

M = cd mod n

Let’s do an example to illustrate the instructions listed above (note: we will be using small prime factors because the math is simpler).

Select a p and q that are prime

P = 11

Q = 3

The modulus n is then equal to P×Q = 11×3 = 33

Computing the totient to be equal to (p-1)×(q-1) = (11-1)(3-1) = 20

To select an “e” we must find a number that is coprime to 20

The smallest value that is coprime to 20 is 3 because 3 is the smallest number that cannot divide 20 evenly, so “e” = 3

Now we need to find “d”, d=e^(-1) mod [(p-1)×(q-1)]

Using the Euclidian Algorithm we get d = 7

Now let’s say we want to encrypt the message “4.” To do this we need to know the public key, which in our case is (n=33, e=3). All we have to do is compute:

C = 43 mod 33 = 31

We can pass 31 (c=31) along to the website, which will then decrypt it using the private key (33, 7):

M = 317 mod 33 = 4

Our message has been successfully “passed” from one place to another.

Thoughts

Without the work from previous mathematicians, this process would not be possible. Modular arithmetic plays a crucial role in our everyday lives and we don’t even notice it. I think it’s an amazing mathematical concept and provides a deep insight into the world of number theory. Even today there are computers constantly trying to figure out how to factor large prime numbers without success. I don’t know if RSA encryption will stand the test of time, but for now it’s the best we’ve got.